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Equilibration of non-extensive systems

Equilibration of non-extensive systems. NEBE parton cascade Zeroth law for non-extensive rules Common distribution Extracting temperatures. T. S. Bíró and G. Purcsel MTA KFKI RMKI Budapest. Talk given at Varos Rab, Croatia, Aug.31-Sept.3 2007. Boltzmann – Gibbs: Extensive S(E,V,N)

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Equilibration of non-extensive systems

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  1. Equilibration of non-extensive systems • NEBE parton cascade • Zeroth law for non-extensive rules • Common distribution • Extracting temperatures T. S. Bíró and G. Purcsel MTA KFKI RMKI Budapest Talk given at Varos Rab, Croatia, Aug.31-Sept.3 2007

  2. Boltzmann – Gibbs: Extensive S(E,V,N) 0: an absolute temperature exists 1: energy is conserved 2: entropy does not decrease spontan. Tsallis and similar: non-extensive 0: ??? 1: (quasi) energy is conserved 2: entropy does not decrease Thermodynamics

  3. NEBE parton cascade Boltzmann equation: Special case: E=|p|

  4. Energy composition rule Associative rule  mapping to addition: quasi-energy Taylor expansion for small x,y and h

  5. Stationary distribution in NEBE Gibbs of the additive quasi-energy = Tsallis of energy Boltzmann-Gibbs in X(E) Generic rule Quasi-energy Tsallis distribution

  6. Abilities of NEBE • Tsallis distribution from any initial distribution • Extensiv (Boltzmann-) entropy • Particle collisions in 1, 2 or 3 dimensions • Arbitrary free dispersion relation • Pairing (hadronization) option • Subsystem indexing • Conserved N, X( E ) and P

  7. Boltzmann: energy equilibration

  8. Tsallis: energy equilibration

  9. Boltzmann: distribution equilibration

  10. Tsallis: distribution equilibration

  11. Mixed: distribution equilibration

  12. Mixed: distribution equilibration

  13. Thermodynamics: general case If LHS = RHS thermal equilibrium, if same function: universal temperature

  14. Thermodynamics: normal case If LHS = RHS thermal equilibrium, if same function: universal temperature

  15. Thermodynamics: NEBE case If LHS = RHS thermal equilibrium, if same function: universal temperature

  16. Thermodynamics: Tsallis case Tsallis entropy: S(E1,E2) = S1 + S2 + (q-1) S1 • S2;  Y(S) additiv, Rényi If LHS = RHS thermal equilibrium, if same function: universal temperature

  17. Thermodynamics: NEBE case  = 1 / T in NEBE; the inverse log. slope is linear in the energy

  18. Boltzmann: temperature equilibration T = 0.50 GeV T = 0.32 GeV T = 0.14 GeV

  19. Tsallis: temperature equilibration T=0.16 GeV, q=1.3054 T=0.12 GeV, q=1.2388 T=0.08 GeV, q=1.1648

  20. Summary • NEBE equilibrates non-extensive subsystems • It is thermodynamically consistent • There exists a universal temperature • Not universal but equilibrates: different T and a systems (not different T and q systems: Nauenberg)

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